Dynamics of Spatial Exploitation:
A Metapopulation Approach
James N. Sanchirico and James E. Wilen
June 2000, Revised October 2000 •
Discussion Paper 00–25-REV
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Dynamics of Spatial Exploitation:
A Metapopulation Approach
James N. Sanchirico and James E. Wilen
Abstract
We present a bioeconomic model of a harvesting industry operating over a heterogeneous environment
comprised of discrete biological populations interconnected by dispersal processes. The model
generalizes the H. S. Gordon [1954]/V. Smith [1968] model of open-access rent dissipation by accounting
for intertemporal and spatial “Ricardian” patterns of exploitation. This model yields a simple, but
insightful, framework from which one can investigate factors that contribute to the evolution of resource
exploitation patterns over space and time. For example, we find that exploitation patterns are driven by
biological and fleet dispersal and biological and economic heterogeneity. We conclude that one cannot
really understand the biological processes operating in an exploited system without knowing as much
about the harvesting system as about the biological system.
Key Words:
renewable resources; bioeconomics; spatial modeling; metapopulation;
JEL Classification Numbers: Q22, R19
This paper is forthcoming in Natural Resource Modeling.
ii
Contents
Introduction .......................................................................................................................................... 1
Metapopulation Model......................................................................................................................... 3
A Model of Spatial Exploitation.......................................................................................................... 4
Spatial Bioeconomic System................................................................................................................ 5
A Spatial and Intertemporal Example.............................................................................................. 6
Comparisons of the Equilibrium Distributions ................................................................................ 8
Comparisons of Spatial and Intertemporal Dynamics .................................................................. 10
Discussion............................................................................................................................................ 17
References ........................................................................................................................................... 19
iii
Dynamics of Spatial Exploitation:
A Metapopulation Approach
James N. Sanchirico and James E. Wilen∗
Introduction
In this paper we present a bioeconomic model of a harvesting industry operating over a patchy
environment comprised of discrete biological populations interconnected by dispersal processes. The
model generalizes the H. S. Gordon [1954] and V. Smith [1968] models of open-access rent dissipation by
accounting for intertemporal and spatial patterns of exploitation. The spatial patterns exhibit the
fundamental process underlying Ricardo's theory of resource scarcity, whereby lands of higher fertility
are brought into cultivation first, followed by the cultivation of successively less profitable land (Barnett
and Morse [1963]). The model of harvester behavior is combined with a metapopulation model that
incorporates modern biological concepts depicting resource patchiness, heterogeneity, and
interconnections among and between patches. This approach yields a simple, but insightful, bioeconomic
model from which one can investigate factors that contribute to the evolution of resource exploitation
patterns over space and time.
Traditional lumped parameter bioeconomic models of fisheries typically begin by assuming a
homogeneous distribution of fish. A biological model is then coupled to a model of a harvesting industry
characterized as operating either as a sole owner without monopoly power or as open access (see, for
example, Clark [1990]). The models of a sole owner illustrate optimal harvest policies and are often
compared to the case of open access to illustrate the inefficiencies associated with lack of ownership first
discussed by H. S. Gordon [1954]. Gordon showed that with the absence of resource ownership, rents in
the fishery would be captured by entrants, and excess returns would attract effort until they were
dissipated. Although Gordon focused on the equilibrium, he acknowledged that the issues affecting
fishery management are inherently a dynamic bioeconomic phenomenon. Vernon Smith, in 1968,
generalized the Gordon model to account for the stock and industry dynamics in a formulation symmetric to
the Lotka-Volterra predator-prey model.
Although the predictions of the Gordon and Smith models are prevalent throughout the resource
economics literature, there is only a small body of empirical literature testing the implications of Gordon's
∗
The authors are Fellow, Quality of the Environment Division, Resources for the Future, and Professor, Department
of Agricultural and Resource Economics, University of California, Davis, respectively. We thank participants at
the Association of Environmental and Resource Economists Workshop (Annapolis, MD, June 1997), and at
seminars at U.C. Davis, Oregon State, Resources for the Future, and North Carolina State for helpful comments.
This research was partially funded by Resources for the Future; by a grant from the National Sea Grant College
Program, NOAA, U.S. Dept. of Commerce, under Grant No. NA36RG0537, Project No. 89-F-N; and by the
Division of Agricultural and Natural Resources and the Giannini Foundation, University of California.
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equilibrium results or the dynamic approach paths highlighted in the V. Smith paper. A paper by Hilborn
and Kennedy [1992] and a paper by Gillis et al. [1993] test and find support for a version of Gordon's rent
equalization hypothesis in the Tasmanian rock lobster fishery and bottom trawl fishery in the Hecate
Strait, respectively. With respect to the transitional dynamics, there are two studies, one of the North
Pacific fur seal industry between 1880 and 1911 (Wilen [1976]) and another of the Northwest Atlantic
harp seal industry (Conrad and Bjorndal [1991]), that showed convergent oscillations of the type
predicted by the Gordon/Smith model.
To illustrate the implications of combining the Gordon/Smith and Ricardian hypotheses within a
bioeconomic model of a metapopulation, we consider the evolution of a fishery from a relatively
unexploited state to an open-access equilibrium. Initially, following Gordon/Smith, one might expect
vessels to enter the fishery when the rents become greater than the opportunity costs of harvest capital. In
addition, following Ricardo, the initial effort level would be concentrated in the most profitable grounds
(or patches), only diffusing into the less profitable areas after the most profitable areas are exploited.
However, unlike the irreversible process imagined by Ricardo, where a land's fertility is depleted and the
farmer moves on to another parcel, renewable resources can rebound from periods of overexploitation in a
metapopulation. Thus the equilibration process might involve both spatial and intertemporal cycles of
over- and undershoot as harvesting effort, harvests, and biomass respond to variations in the patterns of
economic opportunities.
There are many reasons why resource economists might want to investigate the implications of
incorporating the spatial dimension in bioeconomic models.1 First, conservation biologists have long
since abandoned the assumptions of homogeneous environments and distributions of stocks for a new
paradigm that focuses on patchy heterogeneous environments and linkages between the patches. Second,
if a resource is truly distributed heterogeneously in space, then resource economists are most likely
missing a considerable amount of interesting behavior and information by ignoring the spatial dimension.
For example, spatial patterns of vessel and biomass movements across the system could reveal
information about the structural composition of the fishery that could be used to improve resource
stewardship. In addition to these potential modeling gains, spatial models of populations, bioregions, and
ecosystems are being used within the conservation biology literature to promote new marine resource
stewardship concepts referred to as spatial management. For example, biologists have recently promoted
1
In general, ecologists have paid more attention to spatial exploitation patterns than economists have. Hilborn and
Walters [1987] generalized a model used by Caddy [1975] to describe scallop harvesting in a discrete formulation.
Allen and McGlade [1987] developed a multispecies model with a more elaborate depiction of information, and
Klieber and Edwards [1988] developed a simulation model of a spatially mobile tuna/dolphin fishery. See also
McGlade and Allen [1984]; Hilborn and Ledbetter [1979]; MacCall [1990]; Tuck and Possingham [1994];
Supriatna and Possingham [1999]; Martel, Walters, and Wallace [2000], and Walters [2000]. Only recently have
economists begun to focus on the spatial implications of resources exploitation. For example, see a recent paper by
Brown and Roughgarden [1997] examining larval pools in a metapopulation model, papers by Huffaker, Bhat and
Lenhart[1992] and Bhat, Huffaker and Lenhart [1993, 1996] examining spatial/intertemporal control of a pest
population, papers by Skonhoft and Solstad [1996] and Schulz and Skonhoft [1996] analyzing exploitation of
transboundary terrestrial species. See also Bockstael [1996], Albers [1996], Sanchirico [1998], Bulte and Cornelis
[1999], Wilson et al. [1999], and Sanchirico and Wilen [1999].
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Sanchirico and Wilen
natural refuges as management tools, under the expectation that permanent closures of certain areas might
enhance overall biological productivity of an exploited system (Roberts and Polunin [1991]). Similarly,
others have proposed rotating harvest zones, under which areas are closed for a period and then reopened
for exploitation after they have been allowed to recover to a natural state (Botsford, et al. [1993]). To
analyze the bioeconomic implication of these new options, we need a conceptual model that explicitly
considers spatial characteristics of the resource base as well as spatial dimensions of the exploiting
industry.
In the next section, we begin by outlining a biological model of a spatially interconnected population
system and then combine it with a simple open-access harvesting model that captures essential features of
the Gordon/Smith rent dissipation model in a patchy environment. We then solve for the rent-dissipating
equilibrium of the spatially explicit bioeconomic model and present an example to highlight the
implications of spatial heterogeneity on the equilibrium distribution and the adjustment paths of effort and
biomass. We conclude with a discussion of some of the important insights that emerge when we add the
spatial dimension to bioeconomic models of exploitation.
Metapopulation Model
Dating back to the work of Skellam in 1951, biologists and ecologists have been developing
comprehensive and rigorous new theories on the implications of space in models of populations. This
research and modeling effort has culminated in a new class of population models often referred to as
metapopulation models. Metapopulation models are composed of a group of linked subpopulations
distributed across a set of spatially delineated habitats or patches. For the purpose of this paper we
assume that “patches” are locations in space that contain or have the potential to contain an aggregation of
biomass and are located a fixed and discrete distance from one another. The number of organisms in each
patch can be assumed to depend either upon both density-dependent growth processes and dispersal from
and to other patches in the system or only upon growth processes or dispersal. The dispersal process
allows for the possibility of temporary local extinction without driving the whole population to extinction.
This could occur, for example, if the population in one patch is temporarily extinguished and
subsequently recolonized via dispersal from the other patches.
We consider here a general N patch discrete metapopulation model. Following Levin [1974; 1976],
Hastings [1982; 1983], and Vance [1984], let the rate of change of biomass equations across the system
be given by:
!
!
!
" F(X)X+DX
(1)
X=
!
!
" is an n × 1 vector of patch growth rates X
" and X is an n × 1 vector of patch biomass levels
where X
i
Xi. F(X) is a n × n diagonal matrix of own-patch per capita growth functions (Fii=fi(Xi) for all i=1,…n),
which in the case of logistic growth are fi(Xi)=ri(1-Xi/ki). D is an n × n matrix of dispersal coefficients,
where dii is the rate of emigration from patch i (dii<0) and dij is the dispersal rate between patches i and j.2
2Most
models impose some structure on the dispersal process. In this paper we will impose the following restrictions
on the D matrix: (i) dii≤0, (ii) dij≥0, which allow us to identify population fractions that leave from and arrive to one
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The dispersal mechanism in the system could either be density dependent (biomass moves from areas of
higher to lower concentrations) or independent (typically unidirectional flow from a source to a sink
patch(es)).3
This model is capable of depicting a variety of behavioral characteristics of a population and also
oceanographic features of a spatial setting. Different circumstances are best viewed as alternative
parameter restrictions on the dispersal matrix. For example, a closed system is a set of “island” patches,
depicted by D being a null matrix. At the other extreme, a fully integrated system allows for dispersal in a
density-dependent manner from any patch to any other throughout the system and is characterized by a
dispersal matrix with full rank. Other cases would be variants, including linear cascade systems with
linkages only between neighboring patches (D band diagonal), sink-source systems in which one patch
provides unreciprocated biomass replenishment to other patches (Pulliam [1988]; Tuck and Possingham
[1994]) (D a column matrix), or multiple-source cases in which many patches contribute biomass to one
common pool that then is redistributed among the patches.
A discrete model of this type can also depict a range of productivity assumptions in a system of individual
patches (Carr and Reed [1993], Allison et al. [1998], Sanchirico [1998], and Sanchirico and Wilen
[1999]). Some patches may have high biological productivity compared with others, whereas some may
have no inherent productivity, as would be the case with a larval pool that receives and disperses larvae
from a number of other patches. Thus, even in its linear and separable formulation, equation (1) is
relatively general and capable of capturing a broad range of ecological circumstances.
A Model of Spatial Exploitation
In a patchy environment, we might observe an exploiting industry non-uniformly distributed over a finite
number of patches, and the distribution would change over time. Although there is little explicit microtheory of how and why vessels might choose to participate and locate in a particular area, there has been
some empirical work investigating the behavioral motivations for vessel movement patterns across fishing
grounds (patches). Hilborn and Ledbetter [1979] tested whether boats move in traditional patterns, to
maximize weekly catch, or to optimize economic gain subject to the cost of the areas in the British
Columbia salmon purse seine fleet and found strong evidence that vessels moved in an attempt to
optimize economic gain. Eales and Wilen [1986] also found in the California pink shrimp fishery that
patch economic rents were a significant predictor of vessel locational choice decisions. More recently,
Evans [1995] found significant evidence that vessels move to optimize economic gain in the California
area for another. In addition, we assume (iii)
"
n
k !1
d ki ! 0 i=1,2,...,n (column sums to zero), which ensures that
whatever leaves a patch during dispersal from a group of patches also shows up in the receptor patches. A stronger
version of this adding-up restriction is the symmetry condition dij=dji, which ensures that whatever leaves patch i
specifically for j also arrives in j specifically from i. Because this symmetry condition excludes sink-source
dispersal processes, we impose the weaker condition (iii).
3
Density-dependent dispersal processes are employed in papers by Huffaker et al. [1992] and Bhat et al. [1993 and
1996] examining spatial/intertemporal control of a pest population, in papers by Skonhoft and Solstad[1996] and
Schulz and Skonhoft[ 1996] analyzing exploitation of transboundary terrestrial species and in papers by Sanchirico
and Wilen [forthcoming] examining the implications of marine reserve creation.
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Sanchirico and Wilen
salmon fleet. Thus a sensible hypothesis is that the industry responds to relative and absolute rents across
time and space where the differences in rents may be due to economic, biological, and geographical
heterogeneity.
How could such behavior be formally represented? As it turns out, it is relatively straightforward to add a
spatial dispersal component to the Gordon/ Smith model, in a manner similar to the metapopulation
depiction of biological dispersal. Let Ei and Xi denote the patch-specific levels of aggregate effort and
biomass, respectively, in each patch i, and let NRi(Ei,Xi) be the corresponding net rents in patch i. Net
rents are assumed to be average gross operating profits per vessel, less an opportunity cost π per vessel.
Gross operating profits are assumed to be a function of Ei(t) and Xi(t) via a harvesting function Hi(Ei,Xi),
a cost function Ci(Ei,Xi), and a parametric output price p. Opportunity costs per vessel π are assumed to
reflect alternative income earning opportunities outside of the fishery, which we will assume to be
constant per unit of vessel capacity and common across all patches. Thus we can write net rents in patch i
as: NRi=[pH(Ei,Xi)-C(Ei,Xi)-π(Ei)]/Ei.
We then hypothesize that the level of effort, Ei in patch i, will change according to:
N
E" i =si NR i (E i ,Xi )+" sij[NR i (E i ,X i )-NR j (E j ,X j )] for all i=1,...,N
(2)
j=1
j¹i
In this specification, effort in patch i changes in response to two forces. The first is the patch-specific
level of rents vis-à-vis outside opportunities, captured in the first term. When net revenues in patch i
exceed the opportunity cost of vessels, entry occurs from the outside pool of potential effort, and “ownpatch” responsiveness is determined by the rate parameter si. The second force operating on each patch
may be called Ricardian dispersal, depicted by the second term consisting of a sum of pair-wise spatial
dispersal rates, each proportional to rent differentials across space. There will be dispersal from patch j
into patch i if rents in i exceed those in j and dispersal to j from i if the net difference is negative. At any
point in time, patch i may be contributing to a subset of patches experiencing higher relative rents and
drawing from another subset experiencing relatively lower rents. For the system as a whole, these spatial
forces drive the redistribution of effort over space in a manner that equalizes net rents across all patches in
the long run. Because this model assumes that participants are myopic rather than forward-looking
optimizers, an assumption appropriate to the open-access nature of the exploitation depicted, the level and
spatial distribution of the effort operating in the system is not optimal.
Spatial Bioeconomic System
The bioeconomic system depicted here characterizes a harvesting industry behaviorally responsive to
rents within the system vis-à-vis outside opportunities and across opportunities over space. It nests the
Gordon/Smith model when the number of patches is 1 (dij=0 and sij=0). In a manner similar to the
biological system, we can stack equation (2) for all N patches and combine that with the biological system
to get:
! !
" SR(E,X)
E=
(3)
!
!
! !
" F(X)X+DX-H(E,X)
X=
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Sanchirico and Wilen
!
!
n
Here E" and R( E , X ) are n × 1 vectors, S is an n × n matrix ( Sii ! si $ " j !1,i # j sij , and Sij=-sij for
!
i,j=1,…,n with i ≠ j), and H is an n × 1 vector of harvest rates dependent upon both biomass and effort.
In disequilibrium, biological and economic forces are acting in accordance with and against one another
to influence convergence to the bioeconomic equilibrium. For example, in a density-dependent dispersal
process, the high-profit areas will attract larger amounts of effort from the outside pool and from other
patches in the system. As a result of the higher levels of exploitation, high-profit patch biomass levels
drop faster than those in low-profit areas. At the same time, areas with relatively low fishing pressure
initially act as de facto sources for the high-profit patches via dispersal, and this will, other things equal,
speed the convergence of the system to the long-run equilibrium as biomass levels are averaged
throughout the system. In the long run, the system equilibrium can be characterized by:
! set
! !
" = 0 % [F(X)+D]X-H(E,X)=0
X
! set
!
E" = 0 % R(E,X)=0
(4)
!
Note that the matrix of biological dispersal coefficients (D) affects the equilibrium vector of biomass ( X )
!
and effort levels ( E ) in each patch, but the matrix of economic response parameters (S) only affects the
rate of convergence to equilibrium. This occurs because the zero-rent conditions are independent of the
response rates, as in the Vernon Smith model of a single patch. In equilibrium, although the biomass
levels in each patch are constant, the levels in each patch will be maintained in part by biological
dispersal, and hence there will be some biomass movement across space matching outflows to inflows.
Note also that the equilibrium is, in general, fully integrated and simultaneous so that the equilibrium
levels of biomass and effort in each patch depend upon biological and economic parameters (except
response rates) in all other patches. In addition, the character of the equilibrium depends importantly on
the structure of the biological dispersal matrix D.
A Spatial and Intertemporal Example
In this section, we illustrate with an example how dispersal linkages affect both the equilibrium and
intertemporal distribution of effort and biomass throughout the system. As is obvious from the above
description of the biology and economics, there are numerous types of spatial configurations that we
might observe in nature, each with its own implications for the equilibrium and approach paths in a
bioeconomic system. To isolate the implications of the different ecological structures and biological
dispersal mechanisms, we focus on a system in which the fleet dynamics are fully integrated and
represented by the general dispersal system in equations (3) and (4) above. For the biological system, we
consider and compare here i) a closed system, ii) a fully integrated system, and iii) a cascade system. As
it turns out, the closed case is a good benchmark from which to analyze how linkages and densitydependent dispersal affect the distributions over time and space. Note that the three cases can be thought
of as a continuum representing the degree of linkage present in a system, with the closed and fully
integrated systems denoting none and complete linkage, respectively, and the linear or cascade system
falling in between.
For simplicity, we employ a standard set of functional assumptions and a set of rather simple assumptions
regarding the degree of heterogeneity in the fishery. Both sets of assumptions are summarized in Table 1.
With respect to the degree of heterogeneity in the system, we assume that the system is essentially
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Sanchirico and Wilen
homogeneous except for (arbitrarily) assumed differences in costs perhaps due to strong currents or other
oceanographic conditions. In addition, we normalize and rescale the parameters (qi, ri) so that population
is measured in biomass density (xi=Xi/k). Note that by assuming carrying capacities are equal, the
biomass density level is equivalent to the level of biomass in the patches.
Table 1: Function and parameter assumptions.
Functions
Parameter
Restrictions
Harvest
Hi(Ei,xi)= qiEixi
Catchability coefficient
qi=q ∀i
Costs
Ci(Ei,xi)=(ci+πi) Ei
Cost per unit of effort
c1=c2=c, c3=λc, λ>1
Opportunity cost
πi=π ∀i
Prices
pi=p ∀i
Entry/exit response rates
si=s ∀i
Spatial response rates
sij=sji ∀i,j
Intrinsic growth rates
ri=r ∀i
Carrying capacities
ki=k ∀i
Dispersal rates
dii=–Ni*b and dij=b
Economic
Net rents
NRi(Ei,xi)=[piqixi-(ci+πi)]Ei
Biological
Own patch
Production
F(xi) = ri xi (1-xi/ki)
Density-dependent
dispersal
d11x1+d12x2+d13x3=
b(x2/k2-x1/k1)+
Ni = # of patches
connected to patch i
b(x3/k3-x1/k1)
Under these functional assumptions, it is possible to derive a closed-form steady-state solution for the
model. As a first step, we set net rents in patch i equal to zero (NRi = pqixiEi-(ci+π)Ei=0) and solve for xi,
yielding a rent-dissipating equilibrium biomass level for each patch that depends only upon own-patch
economic parameters (xssi≡ (ci+πi)/pqi). As a second step, the biomass levels can then be substituted back
into the biomass equilibrium equations, and the results are closed-form solutions for both biomass and
effort levels. Because net rents are multiplicatively separable with respect to effort, with each patch’s
biomass level dependent only on its own bioeconomic parameters, the biomass levels do not change
across the three cases. A more general model, with rents nonlinear in effort, would integrate our
decoupled special case and make it necessary to solve for the equilibrium values of Ei and xi
simultaneously (Sanchirico [1998]).
Under Table 1 assumptions, we can express the equilibrium biomass levels in terms of the assumed costto-price ratio differences. For example, if x is the population density satisfying rent dissipation in patches
one and two, the density in patch three will be λ x with λ>1, reflecting the fact that patch three is the
high-cost patch
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Sanchirico and Wilen
Comparisons of the Equilibrium Distributions
We first investigate how the spatial equilibrium pattern of effort depends upon the type of biological
system the industry is exploiting. The equilibrium levels of effort and aggregate effort levels across the
three cases appear in Table 2. Note first that the closed system is simply three separate and unconnected
patches and that the equilibrium biomass and effort levels reflect the inherent heterogeneity of the
economic conditions in the system; in particular, where costs are relatively high, the biomass level is
correspondingly high, and effort level is low. Thus the equilibrium density of biomass will be higher in
patch three compared with the two lower-cost patches one and two because fewer vessels are attracted
there because of the higher costs.
Table 2: Steady-state biomass density and effort levels for the
different types of spatial structures.
SPATIAL
STRUCTURE
PATCH EFFORT LEVELS
Closed
E1 ! E2 !
ss
ss
E3 !
ss
Fully-integrated
"
3
i !1
Ei )
r
[2(1 ' x ) $ (1 ' &x )]
q
r
(1 ' x )
q
r
(1 ' & x )
q
r
b
r
2b
[(1 ' x ) ' (1 ' & )] [2(1 ' x ) $ (1 ' &x )] ' [2 ' & ' 1 ]
&
q
r
q
q
r
2b
! [(1 ' &x ) ' (1 ' 1 & )]
q
r
E1 ! E 2 !
ss
E 3 ss
Linear cascade
system
TOTAL EFFORT LEVEL (
ss
r
(1 ' x )
q
r
b
! [(1 ' x ) ' (1 ' & )]
q
r
r
b
! [(1 ' &x ) ' (1 ' 1 & )]
q
r
E1 ss !
(2 is center)
E 2 ss
E 3 ss
r
b
[2(1 ' x ) $ (1 ' &x )] ' [2 ' & ' 1 & ]
q
q
Now what happens to the distribution of effort and biomass as the closed system is “opened up” via
biological linkages? In the fully integrated case with density-dependent dispersal, we find that the spatial
distribution of effort in equilibrium involves more effort in patches one and two and less in patch three
relative to the closed case. This happens because with density-dependent dispersal, the equilibrium costinduced differential in density levels between the patches creates biological dispersal from patch three to
patches two and one. This inflow of biomass from patch three into patches one and two supports larger
amounts of effort in these patches compared with patch three, which can only support lower levels of
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Sanchirico and Wilen
equilibrium effort. Thus, in this case at least, we find that the equilibrium distribution of effort is more
spatially skewed than in the closed case (as is illustrated in Figure 1, Panel A).
Can we say anything about the level of aggregate effort? As it turns out, under these assumptions, the
fully integrated system can support higher levels of effort than the closed case. This occurs because
biomass flows from areas of low profitability (patch three) to areas of high profitability (patches one and
two). Thus the biological dispersal gradient reinforces the economic gradient, creating more potential
rents than in the decoupled system and, as a result, drawing and supporting higher levels of aggregate
effort.
Consider next the linear cascade case, where the patches are located along a line and the center patch is
connected to both outside patches whereas each outside patch is only connected to the center patch. This
case differs from the fully integrated case because the two outside patches are not directly connected to
each other, and hence it illustrates the importance of “edge effects” in a spatial system. In this case, again
the higher population density in patch three generates dispersal from patch three, which can only flow
into patch two. Because patches one and two have identical cost-price ratios, they also have identical
population densities, and hence there is no dispersal from patch two to patch one in equilibrium. Table 2
shows aggregate equilibrium effort levels for patches one, two, and three for these three biological
systems. As can be seen, aggregate equilibrium levels of effort may be equal to, greater than, and less
than corresponding levels in the closed case, respectively. Total effort in the cascade system is greater
than the level in the closed case, for reasons discussed above, but not as great as in the fully integrated
case. This is a spatial Le Chaterlier effect, because the cascade system is more “constrained” by fewer
between-patch linkages than the fully integrated case.4 This implies that potential rents are not as high for
the cascade system, and hence a lower amount of aggregate effort is attracted.
Figure 1: Impact of biological linkages on the spatial distribution of effort
Panel A: Fully-Integrated
Panel B: Linear Cascade
4
The Le Chaterlier Effect was first introduced into economics by Paul Samuelson (1947). The idea was borrowed
from thermodynamics, and the essential point is that the responsiveness of constrained systems to perturbations will
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This simple example hints at some of the richness that emerges when we explicitly consider spatial
factors in a model of bioeconomic exploitation. In this example, the spatial pattern of effort is driven
fundamentally by patch-specific cost-price ratios, but biological spatial linkages and the mechanisms also
influence the equilibrium pattern. We summarize the main results on the equilibrium distribution of
biomass and effort as follows:
•
With a closed structure, the spatial equilibrium of both biomass and effort will reflect the
essential heterogeneity in the biological and economic parameters. For example, if all
patches have the same biological parameters but different economic parameters, high
cost/price patches will have higher biomass densities and lower amounts of effort, other
things equal.
•
When biological linkages connect patches in a system, dispersal of biomass and effort acts to
either homogenize or amplify the density differentials that would exist under a closed system.
•
In a linked system, the degree to which the ultimate spatial equilibrium reflects the
heterogeneity of the fundamental bioeconomic parameters depends upon the biological and
economic gradients. Spatial biological forces can operate in concert with, or against, spatial
economic forces and the equilibrium depends upon the respective strengths and spatial
directions of the two forces.
Comparisons of Spatial and Intertemporal Dynamics
In this section, we investigate the implications of the biological and economic spatial factors on the
transitional dynamics. To do this, we numerically calculate the trajectories for the three-patch system
using solvers specifically designed for nonlinear large-scale stiff ordinary differential equations
(Shampine and Reichelt [1996]). We continue to use the closed case as a benchmark to which to
compare the adjustment paths of the fully integrated and cascade systems. To analyze the effects of the
dispersal/linkages on the dynamics of adjustment, we must specify numerical parameter values. On the
biological side, we continue to assume that the patches are homogeneous with equal intrinsic growth rates
(r=0.8) and carrying capacities (k=10), and a common dispersal rate proportional to the intrinsic growth
rate (b=0.25r). We also assume identical catchability coefficients (q=0.5), prices (p=25), and opportunity
costs (π=3), with patch three harvest costs assumed to be higher than those for patches one and two (c=3,
c3=6 (λ=2)). Finally, we assume that the fleet’s marginal adjustment rates to spatial rents are equal across
the system (sij=s*=0.0004 for all i,j where i≠j) and that the entry response parameters are also equal across
the patches (si=s=0.0055).
In the examples to follow, we illustrate the transitional paths of effort and biomass in each patch, the net
movements of effort between patches, and the net biomass dispersal functions for each patch from a
relatively “unexploited” state (xi(0)=.5ki and Ei(0)=.10* [(ri/qi)*(1-wi/ki)] for i=1,2,3) to an open-access
equilibrium (see Table 2 for equilibrium levels). The net movement of effort for patch i in any period is
defined as: NMi(t)=sij(NRi(t)-NRj(t))+sik(NRi(t)-NRk(t)), where NMi(t)>0 implies movement into patch i
and NMi(t)<0 implies movement out of patch i. Similarly, the general net biomass dispersal function is:
be smaller than the responsiveness of unconstrained systems. In the modeling presented here, the fully integrated
system is less constrained than (for example) the linear cascade system.
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NDi(t)=diixi(t)+dijxj(t)+dikxk(t). If the net biomass dispersal function is positive, then biomass is entering
patch i from the rest of the system, and if negative, then patch i is acting as a source for the system in that
period. Recall that, in equilibrium, net rents are dissipated, implying that the movement of effort both in
and out of the fishery and among the patches goes to zero.
The simulations illustrated here are not policy simulations for a particular fishery, nor are the parameters
calibrated or estimated using fisheries data. Instead, the simulations are intended to illustrate the
possible qualitative effects of economic and biological linkages as determinants of the adjustment to
equilibrium. Although this system is reasonably complex and flexible, patterns do emerge that reveal
how vessels and populations might co-evolve in a heterogeneous spatial environment (Sanchirico [1998]).
The spatial dynamics of the adjustment period for the closed biological system are presented in a series of
panels labeled Figure 2. Panels A and B show individual effort/biomass dynamics in each patch. When
the fishing pressure begins to increase, effort is drawn into all three patches from an outside pool. Then
as biomass is driven down, profits turn negative and system-wide exit occurs. This in turn reduces
harvest rates until biomass begins to recover and the second cycle of the convergent oscillatory process
begins. In the first phase, effort levels in patches one and two are larger than those in patch three because
of higher initial profits. The greater effort levels drive patches one and two to lower stock levels than in
patch three. As a consequence, patch three has higher relative net rents and ends up attracting some of the
vessels exiting patches one and two (see Figure 3). These cycles of overshoot and undershoot repeat and
dampen as the system approaches equilibrium, in which patch three ends up with higher biomass and
lower effort than patches one and two because of the assumed differences in costs.
From a whole-system perspective, the patchy model reproduces the results that Vernon Smith [1968;
1969] deduced in his simple single-patch model. However, there are important patch-specific differences
in behavior that result in oscillations in the spatial distribution of effort that are not synchronized. For
example, although each patch exhibits over- and undershoot, this occurs as a result of the fleet’s
movement among patches as well as from inflow and outflow to the outside pool. Figure 2 panel C plots
the vessels’ movements between patches in response to relative rents. The different qualitative spatial
cycles illustrated in Figure 2 panel C are decomposed and summarized in Figure 3.5 In the first cycle of
overshoot, patches one and two are relatively more profitable to operate in, due to the variable operating
cost assumptions, than patch three, thereby attracting effort from patch three. The combination of
attracting effort from patch three simultaneously with the entry from an outside pool of effort compound
one another, and, consequently, the level of effort operating in patches one and two increases at faster rate
than in patch three. The increase in effort levels is transmitted through the harvest production function
into biomass levels, and, consequently, the level of biomass in patch three
5
In any period the amount of effort leaving patch i for patch j is: sij(NRi-NRj), where NRj>NRi. For illustrative
purposes, the adjustment period is broken into different phases of vessel movement patterns in which the phases are
distinguished by qualitatively different spatial movement. For example, suppose a patch is attracting vessels over
one period and then the patch begins to repel vessels over another. In this case, we would characterize the
movement pattern with two phases. This means that the phases in one example do not coincide with those in other
examples, although the graphs of the levels over time can be used to compare the differences in timing between the
phases.
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Sanchirico and Wilen
Figure 2: Adjustment paths in a closed biological system
Panel B: Biomass Levels
Panel A: Effort LeveLs
6
2.5
Biomass
Effort
2
1.5
4
2
1
0
0.5
0
200
400
600
0
800
1
0.01
0.5
Movements
Movements
0.02
0
-0.01
200
400
600
800
Time
Panel D: Net Biological Dispersal
Time
Panel C: Net Effort Movements
0
-0.5
-1
-0.02
0
200
400
600
0
800
Time
200
400
600
800
Time
*The solid line represents patches one and two and the dashed line represents patch three.
falls at a slower rate than in patches one and two. In the second phase, patch three, which has remained
relatively free from harvest, begins to attract effort from patches one and two. Overall, the spatial
movements of effort follow a rotating harvest pattern where vessels fish down a patch and then move on
to other areas. This provides an opportunity (albeit temporary) for biomass levels to recover from periods
of heavy exploitation. In the end, however, these phases dampen in the approach to equilibrium, at
which point rents are equalized and dissipated and movements both in and out and across the (closed)
system are zero.
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Figure 3: Between-patch fleet dynamics during over- and undershoot phases.
P h a se I
P h a se II
We investigate next, as we did in the previous section, the effects of biological linkages. In general, in
the fully integrated case, the within-patch cyclical patterns of effort, biomass, and harvest levels are
qualitatively similar to those in the closed case. Because of these similarities, we focus here only on the
points of departure between the two cases. The first and rather obvious point is that in the fully integrated
case there exists biological dispersal between the patches during the approach to equilibrium. Figure 4,
panel B, illustrates the manner in which a patch is a net receptor or contributor of biomass in any period,
and Figure 5 decomposes this into the individual patch interactions throughout the adjustment period. For
example, in the first phase, we find that patch three acts as a temporary de facto source of biomass for
both patches one and two because of the lower initial rents from the higher costs per unit of effort.
However, as vessels move to patch three, the biomass levels in patches one and two begin to recover. As
the biomass recovers in patches one and two, the level in patch three is dropping because of higher effort
levels. Consequently, in the second phase patches one and two reverse roles and begin to act as de facto
source patches for patch three. As can be seen in Figure 4, panel B, this reversal is short lived, and in the
remainder of the adjustment period patch three returns to acting as the de facto source for the system
(phase A in Figure 5).
Aside from the obvious effects of biomass dispersal, there are two rather subtle yet fundamental
differences between the closed and fully integrated case. First, the magnitudes of the effort movements
differ, with lower peaks and troughs witnessed at turning points in the integrated system. Second, the
amount of “time” until the system converges to the equilibrium is shorter in the fully integrated than in
the closed case.6 What mechanisms drive these differences? Essentially, the density-dependent
movements of the biomass out of patch three into patches one and two reduce the initial differential
caused by price-cost ratio differences. This, in turn, reduces the magnitude of the effort dispersal needed
to bring net rents into equilibrium. Thus, whereas in the closed case only the fleet can move between
areas to bring the economic system into equilibrium, in the fully integrated system, both biomass and
effort dispersal are acting and operating in ways that reinforce each other in equalizing biomass densities
and rents across the system. This can be seen in Figure 4, panels A and B, where the two processes are
essentially smoothing out differences across the system and bringing the whole system closer to the
neighborhood of equilibrium earlier than would be the case without biological (density-dependent)
dispersal linkages.
6
In this example, the ratio of the length of the adjustment period between the cases (fully integrated/closed) is 0.80.
Also, note that the length of time until rents are dissipated is a function of the industry’s entry/exit response rates
(si), intrinsic growth rates (ri), economic and biological dispersal rates (sij and dij), and initial conditions.
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Figure 4: Movement patterns in the fully integrated case.*
Panel A: Net Effort Movements
Movements
0.01
0.005
0
-0.005
-0.01
0
100
200
300
400
500
600
Time
Panel B: Net Biological Dispersal
Biomass
0.02
0
-0.02
-0.04
0
100
200
300
400
500
600
Time
*The solid line represents patches one and two and the dashed line represents patch three.
Figure 5: Between-patch biomass dynamics during over- and undershoot phases
P h a se A
P h a se B
What are the intertemporal implications of the spatial Le Chaterlier effect? In other words, how does the
approach path pattern in a system that is more constrained (with fewer linkages) compare to the two
extreme cases? The adjustment of the effort and biomass in the linear cascade case is illustrated in
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Figure 6. The patterns of effort entering the fishery are very similar to the previous cases. After the initial
entry phase, however, a divergence between patches one and two emerges that is a result of the biological
“edges” in this system. This is illustrated in Figure 6, panels C and D, and summarized in Figures 7 and
8. It can be seen that the effort and biomass levels in patch two begin to deviate from those in patch one
because of the higher profits in patch two resulting from the dispersal of biomass from the adjacent (and
high cost) patch three. In the fully integrated case, biomass leaving patch three is shared equally between
patches one and two because the system is fully integrated, resulting in identical industry adjustments
between the two patches.
Figure 6: Adjustment paths in the linear cascade system.*
Panel A: Effort Levels
2.5
Panel B: Biomass Levels
6
Biomass
Effort
2
1.5
4
2
1
0.5
0
0
200
400
600
800
0.02
0.02
0.01
0.01
0
-0.01
0
200
400
600
0
200
400
600
800
Time
Panel D: Net Biological Dispersal
Biomass
Movements
Time
Panel C: Net Effort Movements
0
-0.01
-0.02
-0.02
0
200
400
600
800
800
Time
Time
* The solid line represents patch one, the dotted line represents patch two, and the dashed line represents patch three.
Another interesting feature of this system has to do with the spatial characteristics of the fleet’s approach
to equilibrium. In this case, there exist five phases in the adjustment process, as illustrated in Figure 6
and detailed in Figure 8. We find that because of the lack of inflow of biomass via dispersal (Figure 7,
phase A), rents in patch one decrease faster than in patch two while at the same time the fishing pressure
in patch three is relatively low. As a result, vessels begin to leave patch one for patch two and three. In
addition, as a result of the relative lack of exploitation in patch three in phase I, rents are temporarily
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higher than in patches one and two, creating incentives for vessels to move back into patch three. As with
the closed and fully integrated systems, these cycles of overshoot and undershoot repeat (see Figure 6,
panel C) and dampen as the system approaches equilibrium. Unlike the closed and fully integrated
systems, however, the existence of a structural asymmetry results in a more complex pattern of biomass
and effort over time.
Figure 7: Spatial patterns of biomass movements in the linear cascade system
P h a se A
P h a se B
P h a se C
P h a se D
P h a se E
Figure 8: Spatial patterns of fleet movements in the linear cascade system
P h a se I
P h a se II
P h a se III
P h a se IV
P h a se V
Although the exact nature of the overshoot-undershoot cycles within and across patches and the speeds of
adjustments are particular to the set of parameter values and functional forms, the qualitative results from
the numerical analysis highlight the effects of linkages on spatial and intertemporal resource exploitation.
The various cases discussed above illustrate and highlight different characteristics of biological and
economic systems that can be attributed to observed patterns of exploitation. These characteristics are
summarized by the following points:
•
In the closed case, vessel movement is the only endogenously driven mechanism smoothing
out the initial effects of the cost/price heterogeneity and biomass distribution.
•
The fully integrated system illustrates how economic and biological dispersal homogenize
rent and biomass density differences when both the biological and economic gradients act
together to dissipate the dissimilarities across the patches in the system.
•
The spatially linear or cascade case illustrates the effects of biological “edges,” which can act
in complement with or in opposition to the dynamics driven by cost/price and biological
heterogeneities. As a result of the asymmetries introduced by edges, this system exhibits
more complex spatial and intertemporal behavior than the other systems.
The magnitude and length of the adjustment period depends upon the number and type of the economic
and biological linkages in the system. Ceteris paribus, more linkages speed the adjustment process. In
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addition, having a system in which the biological gradient operates in tandem with the economic gradient
speeds adjustment.7
Discussion
The model developed here generalizes traditional bioeconomic models based on assumptions of
homogeneous distributions of biomass across space. The biological model is linked to a spatial model of
harvester behavior that is economically motivated and endogenous to the spatial distribution of the
resource. In this model, rent dissipation is occurring across more than one margin. That is, not only are
rents being dissipated by attracting effort from the outside pool, they are also being dissipated by vessels
moving back and forth across the system.
While the structure of the model is very flexible and easily adaptable to more complex depictions of the
biological and economic systems, we choose to illustrate the model with a very stylized example in order
to highlight the effects of incorporating the spatial dimension into traditional bioeconomic models. In
previous work, we have investigated the implications of more general net rent functions and done
extensive sensitivity analysis on the parameters, and all of the qualitative results still hold (Sanchirico
[1998]). Of course, the rates of convergence and the specific movement patterns of the numerical
analysis are conditional on our functional form, parameter assumptions, and initial conditions. Even
though the qualitative results are in line with the empirical findings, it is unclear if the convergence
properties of this model will be reinforced when tested in more complex biological (e.g., discrete space
and time logistic model, age/size structure, migration patterns) and economic systems (e.g., transportation
costs, informational lags). Whether the functional relationships and relative parameter values used here
are reflective of the marine environment, however, is an empirical question and left for further research.
In general, the incorporation of space into the analysis of resource management leads to important
insights into how and why patterns of resource exploitation might develop across space and time. For
instance, on a fundamental level, biological and economic heterogeneity across space plays a large role in
determining spatial distributions of effort and biomass. However, at the same time, linkages via dispersal
(both fleet and biomass) can either exacerbate or dampen the influences of heterogeneity depending upon
the bioeconomic dispersal mechanisms assumed. In the example presented here, we found that the
distribution is more spatially skewed in the presence of linkages with an endogenous density-dependent
dispersal process than when the system is biologically closed.
The spatial and intertemporal patterns of exploitation generally depict a Ricardian process in which areas
of higher profitability are exploited more intensively earlier. For instance, as the fishery is opened to
exploitation, effort flows into each of the patches at a rate proportional to the net rents in the patch and the
differences in rents across the systems. As the fishery develops, the most profitable patches have vessels
7
For example, we have also analyzed a three-patch sink-source system in which patch three is the source and the
high-cost patch and found that the system converges to an equilibrium relatively fast (Sanchirico [1998]). This is
because biomass dispersal would flow in the same direction as effort needed to flow to bring both biomass and effort
into equilibrium. The same biological system with patch one as the high-cost patch would reach equilibrium more
slowly, because the economic gradient would be “pushing” effort in the direction opposite to that driven by the
biological gradient.
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Sanchirico and Wilen
entering not only from outside the fishery but also from the other, less-profitable areas within the fishery.
As a result of the initial periods of high effort and relative overharvesting, the biomass levels in highprofit areas drop faster than in the other patches, and ultimately it becomes more profitable to move to
other areas. After some time, the biomass in the area that was initially overharvested begins to recover
and vessels return to capture the spatial rents. This process continues in a dampening set of over- and
undershoot cycles until, in equilibrium, rents are dissipated and vessels are no longer moving between
areas.8
An important point highlighted in this paper is that whether or not a patch is acting as a de facto source or
sink depends not only on the biological factors but also on the economic conditions and the particular
phase of the adjustment process. In essence, “sources” and “sinks” in a density-dependent system are
intertemporal bioeconomic phenomenon. We find, for example, that cost differences in a biologically
homogeneous system generate biological dispersal between patches. In this case, conventional markrecapture tagging might conclude that the high-cost patch is a source and could attribute this to unique
biological characteristics when, in fact, biomass movement is fundamentally due to economic factors. In
addition, as our example illustrated, a patch can change from a source to a sink in a system because of
relative biological and economic gradients, an observation that has implications for spatial management.
For example, in choosing a location for a marine reserve, it is possible that what might appear over one
period of time to be an unusually biologically productive patch might actually turn out to be a relatively
unproductive patch after the patch is closed.
Finally, there are number of implications that can be drawn from this modeling exercise regarding the
viability of various spatial management options. First of all, the model suggests that understanding more
about the role space plays in a patchy system of resource exploitation is vital to any policy analysis. Of
particular importance is the conclusion that we cannot really understand the biological processes
operating in an exploited system without knowing as much about the harvesting system as we do about
the biological system. This generally calls for much more emphasis on joint bioeconomic modeling and
investigations of both economic processes operating over space and time and biological processes. With a
better understanding of the implications of space in an exploited system, we can begin to sensibly address
the policy questions concerning how spatial management might affect the overall “health” of the fishery,
how choices based upon biological and economic heterogeneity might be made, and how biological and
economic linkages affect whether or not these policies might prove beneficial.
8 We
focused here on cases where vessels are free to move between all the patches under various ecological
structures from the most restricted (closed) to the least restricted (fully integrated). As a result, the structural
asymmetries are only present in the biological system. However, another question to ask is, how do structural
asymmetries in the fishing sector affect the spatial and intertemporal bioeconomic patterns? These structural
asymmetries in the fishing sector could be the product of the 200-mile exclusive economic zone or institutional
rigidities within a state or nation. For instance, if vessel mobility is restricted to a subset of the biological system of
patches, then it is possible that this structural asymmetry could result in maintaining the effects of spatial
heterogeneity for longer periods of time. These results might be analogous to the closed case above where, instead
of the fleet moving about to equalize net rents, the dispersal of biomass would equalize the patch densities.
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